Wiener空间上的重整化群：谱理论与普适性
Renormalization Group on Wiener Space: Spectral Theory and Universality
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中文总结 AI 辅助
本文在Wiener空间上建立严格重整化群形式体系，通过白噪声分析实现谱理论，揭示Donsker不变原理中修正层级的新普适性，并恢复量子场论中的标准分类。
中文摘要 AI 辅助
我们在Wiener空间上发展了一个严格的重整化群（RG）形式体系。我们引入了一个作用于连续路径上概率测度的RG算子$\mathcal R$，其不动点是Wiener测度。我们在Brownian不动点附近将$\mathcal{R}$线性化，并分析所得算子$\mathscr{L}$的谱结构。其特征向量过于奇异，无法实现为路径空间上的正当测度，因此我们在白噪声分析（Hida微积分）框架内发展了一个广义谱理论。$\mathscr{L}$的特征向量被实现为Hida分布，满足弱意义下的$\mathscr{L} U=\lambda U$。这一分析产生了结构性的结果，例如谱间隙和特征向量的对角集中性质。此外，我们确定了一个特征向量族$\{\mathfrak{U}_n\}_{n \geq 0}$，其特征值为$\lambda_n = 2^{1-n/2}$，由白噪声的Wick多项式组成，形式上表示为$\mathfrak{U}_n = \int_0^1 {:}\dot W(t)^n{:}\\, dt$，并严格构造为Hida分布。然后，我们利用这一谱结构揭示了Donsker不变原理中一个更精细的第二层普适性：不仅随机游走向Brownian标度极限的收敛是普适的，而且对Brownian极限的整个主导修正层级也是普适的，由特征对$(\lambda_n, \mathfrak{U}_n)$所支配。我们还形式化地表明，这里发展的框架可扩展到Brownian不动点的Gibbs型扰动，类似于自相互作用量子场论的精神。特别是，我们仅从$\mathscr{L}$的谱分析中恢复了通常在物理文献中通过幂次计数得到的标准无关/边缘/相关分类。
英文摘要
We develop a rigorous renormalization group (RG) formalism on Wiener space. We introduce an RG operator $\mathcal R$ acting on probability measures over continuous paths, whose fixed point is the Wiener measure. We linearize $\mathcal{R}$ around the Brownian fixed point and analyze the spectral structure of the resulting operator $\mathscr{L}$. Its eigenvectors are too singular to be realized as honest measures on path space, and we therefore develop a generalized spectral theory within the framework of white noise analysis (Hida calculus). The eigenvectors of $\mathscr{L}$ are realized as Hida distributions, satisfying $\mathscr{L} U=λU$ in the weak sense. This analysis yields structural results such as a spectral gap and a diagonal-concentration property of the eigenvectors. Moreover, we identify a family $\{\mathfrak{U}_n\}_{n \geq 0}$ of eigenvectors with eigenvalues $λ_n = 2^{1-n/2}$, consisting of Wick polynomials of white noise formally expressed as $\mathfrak{U}_n = \int_0^1 {:}\dot W(t)^n{:}\, dt$, and rigorously constructed as Hida distributions. We then use this spectral structure to uncover a finer, second layer of universality in the Donsker invariance principle: not only is the convergence of random walks towards the Brownian scaling limit universal, but the entire hierarchy of leading corrections to the Brownian limit is universal as well, governed by the eigenpairs $(λ_n, \mathfrak{U}_n)$. We also show, at a formal level, that the framework developed here extends to Gibbs-type perturbations of the Brownian fixed point, in the spirit of self-interacting quantum field theories. In particular, we recover the standard irrelevant/marginal/relevant classification, usually obtained in the physics literature by power-counting, solely from the spectral analysis of $\mathscr{L}$.
发表机构
- Instituto de Matemática Pura e Aplicada (IMPA)（巴西纯粹与应用数学研究所）
- Université Côte d’Azur（蔚蓝海岸大学）
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